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The remainder term arises because the integral is usually not exactly equal to the sum. The formula may be derived by applying repeated integration by parts to successive intervals [r, r + 1] for r = m, m + 1, …, n − 1. The boundary terms in these integrations lead to the main terms of the formula, and the leftover integrals form the ...
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.
That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be ...
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
It can be seen that as N gets larger (1 + iπ / N ) N approaches a limit of −1. Euler's identity asserts that e i π {\displaystyle e^{i\pi }} is equal to −1. The expression e i π {\displaystyle e^{i\pi }} is a special case of the expression e z {\displaystyle e^{z}} , where z is any complex number .
Sum S of internal angles of a regular convex polygon with n sides: = Area A of a regular convex polygon with n sides and side length s: = Inradius r of a regular convex polygon with n sides and side length s: